We introduce and investigate using Hilbert modules the properties of the Fourier algebra A(G) for... more We introduce and investigate using Hilbert modules the properties of the Fourier algebra A(G) for a locally compact groupoid G. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard.
Proceedings of the American Mathematical Society, Apr 20, 2005
The path spaces of a directed graph play an important role in the study of graph C * -algebras. T... more The path spaces of a directed graph play an important role in the study of graph C * -algebras. These are topological spaces that were originally constructed using groupoid and inverse semigroup techniques. In this paper, we develop a simple, purely topological, approach to this construction, based on Tychonoff's theorem. In fact, the approach is shown to work even for higher dimensional graphs satisfying the finitely aligned condition, and we construct the groupoid of the graph. Motivated by these path space results, we prove a Tychonoff theorem for an infinite, countable product of locally compact spaces. The main idea is to include certain finite products of the spaces along with the infinite product. We show that the topology is, in a reasonable sense, a pointwise topology.
Proceedings of the American Mathematical Society, 2005
The path spaces of a directed graph play an important role in the study of graph C * -algebras. T... more The path spaces of a directed graph play an important role in the study of graph C * -algebras. These are topological spaces that were originally constructed using groupoid and inverse semigroup techniques. In this paper, we develop a simple, purely topological, approach to this construction, based on Tychonoff's theorem. In fact, the approach is shown to work even for higher dimensional graphs satisfying the finitely aligned condition, and we construct the groupoid of the graph. Motivated by these path space results, we prove a Tychonoff theorem for an infinite, countable product of locally compact spaces. The main idea is to include certain finite products of the spaces along with the infinite product. We show that the topology is, in a reasonable sense, a pointwise topology.
The Fourier-Stieltjes and Fourier algebras B(G), A(G) for a general locally compact group G, firs... more The Fourier-Stieltjes and Fourier algebras B(G), A(G) for a general locally compact group G, first studied by P. Eymard, have played an important role in harmonic analysis and in the study of the operator algebras generated by G. Recently, there has been interest in developing versions of these algebras for locally compact groupoids, justification for this being that, just as in the group case, the algebras should play a useful role in the study of groupoid operator algebras. Versions of these algebras for the locally compact groupoid case appear in three related theories: (1) a measured groupoid theory (J. Renault), (2) a Borel theory (A. Ramsay and M. Walter), and (3) a continuous theory (A. Paterson). The present paper is expository in character. For motivational reasons, it starts with a description of the theory of B(G), A(G) in the locally compact group case, before discussing these three theories. Some open questions are also raised.
Let G be an amenable locally compact group. It is conjectured that every topological left invaria... more Let G be an amenable locally compact group. It is conjectured that every topological left invariant mean on Lco(G) is (topologically) invariant if and only if Ge [FC]~. This conjecture is shown to be true when G is discrete and when G is compactly generated.
We introduce and investigate using Hilbert modules the properties of the Fourier algebra A(G) for... more We introduce and investigate using Hilbert modules the properties of the Fourier algebra A(G) for a locally compact groupoid G. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard.
Transactions of the American Mathematical Society, 1997
We investigate the cohomology of non-self-adjoint algebras using virtual diagonals and their high... more We investigate the cohomology of non-self-adjoint algebras using virtual diagonals and their higher-dimensional generalizations. We show that infinite dimensional nest algebras always have non-zero second cohomology by showing that they cannot possess 2-virtual diagonals. In the case of the upper triangular atomic nest algebra we exhibit concrete modules for non-vanishing cohomology.
We develop higher dimensional amenability for Banach algebras from the viewpoint of Banach homolo... more We develop higher dimensional amenability for Banach algebras from the viewpoint of Banach homology theory. In particular, we show that such amenability is equivalent to the flatness of a certain bimodule and a resultant splitting module map gives rise to the higher dimensional virtual diagonals of Effros and Kishimoto. The theory is developed for the nonunit al case. Examples of n-amenability are given and it is shown (among other results) that a 2-amenable Banach algebra is amenable if and only if there exists an inner 2-virtual diagonal. 1. Introduction.
Transactions of the American Mathematical Society, 1991
In this paper we study the relationship between amenability, inner amenability and property P of ... more In this paper we study the relationship between amenability, inner amenability and property P of a von Neumann algebra. We give necessary conditions on a locally compact group G to have an inner invariant mean m suchthat m(V) = 0 for some compact neighborhood F of G invariant under the inner automorphisms. We also give a sufficient condition on G (satisfied by the free group on two generators or an I.C.C. discrete group with Kazhdan's property T, e.g., SL(«, Z), n > 3) such that each linear form on L (G) which is invariant under the inner automorphisms is continuous. A characterization of inner amenability in terms of a fixed point property for left Banach G-modules is also obtained. Amenable locally compact groups and [IN]-groups are inner amenable. The group G is [IN] if there exists a compact neighborhood V of the identity e in G such that a~xVa = V for all a e G. Furthermore when G is connected, then G is amenable if and only if G is inner amenable (see [17]). A recent account of amenability is given in [21]. Let J' be a von Neumann algebra on a Hilbert space H and let Jü' be the commutant of ./#. For T e 38(H) (the space of bounded linear operators on H), let CT be the weak*-closed convex subset of 38'(H) generated by {U*TU ; U e Jfu}, where ^u is the group of unitary elements in Jf. (Note that 38(H) has a unique predual [28, p. 47].) Jf is said to have property P if CT xx Jf' # 0 for each T e 38(H). Let VN((j) denote the von Neumann algebra on L (G) generated by {lx;x e G} where lxh(t) = h(xt), t e G. A well-known result of Schwartz [29] asserts
In his study of amenable unitary representations, M. E. B. Bekka asked if there is an analogue fo... more In his study of amenable unitary representations, M. E. B. Bekka asked if there is an analogue for such representations of the remarkable fixed-point property for amenable groups. In this paper, we prove such a fixed-point theorem in the more general context of a Gamenable von Neumann algebra M , where G is a locally compact group acting on M. The Følner conditions of Connes and Bekka are extended to the case where M is semifinite and admits a faithful, semifinite, normal trace which is invariant under the action of G.
Proceedings of the Edinburgh Mathematical Society, 1989
In [5], Ky Fan proved the following remarkable amenability “invariant subspace” theorem:Let G be ... more In [5], Ky Fan proved the following remarkable amenability “invariant subspace” theorem:Let G be an amenable group of continuous, invertible linear operators acting on a locally convex space E. Let H be a closed subspace of finite codimension n in E and X⊂E be such that:(i) H and X are G-invariant;(ii) (e + H) ∩X is compact convex for all e ∈ E;(iii) X contains an n-dimensional subspace V of E. Then there exists an n-dimensional subspace of E contained in X and invariant under G.
Proceedings of the American Mathematical Society, 1988
Let G G be a locally compact group and p ∈ [ 1 , ∞ ] p \in [1,\infty ] . Let π p {\pi _p} be the ... more Let G G be a locally compact group and p ∈ [ 1 , ∞ ] p \in [1,\infty ] . Let π p {\pi _p} be the isometric representation of G G on L p ( G ) {L_p}(G) given by π p ( x ) f ( t ) = f ( x − 1 t x ) Δ ( x ) 1 / p {\pi _p}(x)f(t) = f({x^{ - 1}}tx)\Delta {(x)^{1/p}} . Let A p ′ {\mathcal {A}’_p} be the commutant of A p {\mathcal {A}_p} in B ( L p ( G ) ) B({L_p}(G)) . In this paper we determine those G G for which: (*) A p ′ {\mathcal {A}’_p} contains a nonzero compact operator. We prove, among other things, that if p ∈ [ 1 , ∞ ) p \in [1,\infty ) , then (*) holds if and only if G ∈ [ I N ] G \in [IN] , and that if p = ∞ p = \infty , then (*) holds if and only if G G is inner amenable.
Proceedings of the American Mathematical Society, 1986
The purpose of this note is to prove that if G G is an amenable locally compact noncompact group,... more The purpose of this note is to prove that if G G is an amenable locally compact noncompact group, then the set of topological left invariant means on L ∞ ( G ) {L_\infty }(G) has cardinality 2 2 d {2^{{2^d}}} , where d d is the smallest cardinality of the covering of G G by compact sets. We also prove that in this case the spectrum of the bounded left uniformly continuous complex-valued functions contains exactly 2 2 d {2^{{2^d}}} minimal closed invariant subsets (or left ideals)
and New South Wales. Responsibility for any errors remains with the author. 2 Institutional indep... more and New South Wales. Responsibility for any errors remains with the author. 2 Institutional independence refers to the autonomy of the Legal Aid Authority from its sponsoring Ministry in terms of its location inside or outside Government. Operational independence refers to the LAA"s autonomy from Government (and other influences) in relation to the granting or refusing of legal aid. Financial independence refers to the LAA"s autonomy from others in relation to its funding. 3 The Irish LAB is technically an agency of the MOJ but it is situated outside the MOJ. 4 LA Ontario is described as an operational agency of the Government of Ontario but it is outside the Government. 5 Described as a Crown Corporation which is not a Government agency. 6 This gives the Government the important protection of deniabilitythe decision was not taken by us, but by an independent body, the LAA.
Of the four Consultation Papers produced by the Department for Constitutional Affairs over the su... more Of the four Consultation Papers produced by the Department for Constitutional Affairs over the summer of 2003, in many ways the most interesting was Constitutional Reform: A Supreme Court for the United Kingdom, not just for what it did say, but also for what it did not. For example, respondents were not asked whether the Government should replace the House of Lords with a Supreme Court or not. That was taken as a given. Yet the omission was all the more curious in the light of the fact, as subsequently became clear, that at least half of the current Law Lords do not favour the introduction of a Supreme Court.
International Journal of the Legal Profession, 1994
The legal aid system in England and Wales is in a stage of rapid change. Faced with a burgeoning ... more The legal aid system in England and Wales is in a stage of rapid change. Faced with a burgeoning budget, the Lord Chancellor's Department has decided to reduce eligibility, create standard fees for criminal cases and set a standard rate for Civil Legal Aid work. The Legal Aid Board's (LAB) new'franchising'scheme also has management of costs as a concern, 2 but the professed aim is broader: it is to ensure a'value for money'service through the accreditation of legal aid practices that meet standards for quality. This presages ...
We introduce and investigate using Hilbert modules the properties of the Fourier algebra A(G) for... more We introduce and investigate using Hilbert modules the properties of the Fourier algebra A(G) for a locally compact groupoid G. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard.
Proceedings of the American Mathematical Society, Apr 20, 2005
The path spaces of a directed graph play an important role in the study of graph C * -algebras. T... more The path spaces of a directed graph play an important role in the study of graph C * -algebras. These are topological spaces that were originally constructed using groupoid and inverse semigroup techniques. In this paper, we develop a simple, purely topological, approach to this construction, based on Tychonoff's theorem. In fact, the approach is shown to work even for higher dimensional graphs satisfying the finitely aligned condition, and we construct the groupoid of the graph. Motivated by these path space results, we prove a Tychonoff theorem for an infinite, countable product of locally compact spaces. The main idea is to include certain finite products of the spaces along with the infinite product. We show that the topology is, in a reasonable sense, a pointwise topology.
Proceedings of the American Mathematical Society, 2005
The path spaces of a directed graph play an important role in the study of graph C * -algebras. T... more The path spaces of a directed graph play an important role in the study of graph C * -algebras. These are topological spaces that were originally constructed using groupoid and inverse semigroup techniques. In this paper, we develop a simple, purely topological, approach to this construction, based on Tychonoff's theorem. In fact, the approach is shown to work even for higher dimensional graphs satisfying the finitely aligned condition, and we construct the groupoid of the graph. Motivated by these path space results, we prove a Tychonoff theorem for an infinite, countable product of locally compact spaces. The main idea is to include certain finite products of the spaces along with the infinite product. We show that the topology is, in a reasonable sense, a pointwise topology.
The Fourier-Stieltjes and Fourier algebras B(G), A(G) for a general locally compact group G, firs... more The Fourier-Stieltjes and Fourier algebras B(G), A(G) for a general locally compact group G, first studied by P. Eymard, have played an important role in harmonic analysis and in the study of the operator algebras generated by G. Recently, there has been interest in developing versions of these algebras for locally compact groupoids, justification for this being that, just as in the group case, the algebras should play a useful role in the study of groupoid operator algebras. Versions of these algebras for the locally compact groupoid case appear in three related theories: (1) a measured groupoid theory (J. Renault), (2) a Borel theory (A. Ramsay and M. Walter), and (3) a continuous theory (A. Paterson). The present paper is expository in character. For motivational reasons, it starts with a description of the theory of B(G), A(G) in the locally compact group case, before discussing these three theories. Some open questions are also raised.
Let G be an amenable locally compact group. It is conjectured that every topological left invaria... more Let G be an amenable locally compact group. It is conjectured that every topological left invariant mean on Lco(G) is (topologically) invariant if and only if Ge [FC]~. This conjecture is shown to be true when G is discrete and when G is compactly generated.
We introduce and investigate using Hilbert modules the properties of the Fourier algebra A(G) for... more We introduce and investigate using Hilbert modules the properties of the Fourier algebra A(G) for a locally compact groupoid G. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard.
Transactions of the American Mathematical Society, 1997
We investigate the cohomology of non-self-adjoint algebras using virtual diagonals and their high... more We investigate the cohomology of non-self-adjoint algebras using virtual diagonals and their higher-dimensional generalizations. We show that infinite dimensional nest algebras always have non-zero second cohomology by showing that they cannot possess 2-virtual diagonals. In the case of the upper triangular atomic nest algebra we exhibit concrete modules for non-vanishing cohomology.
We develop higher dimensional amenability for Banach algebras from the viewpoint of Banach homolo... more We develop higher dimensional amenability for Banach algebras from the viewpoint of Banach homology theory. In particular, we show that such amenability is equivalent to the flatness of a certain bimodule and a resultant splitting module map gives rise to the higher dimensional virtual diagonals of Effros and Kishimoto. The theory is developed for the nonunit al case. Examples of n-amenability are given and it is shown (among other results) that a 2-amenable Banach algebra is amenable if and only if there exists an inner 2-virtual diagonal. 1. Introduction.
Transactions of the American Mathematical Society, 1991
In this paper we study the relationship between amenability, inner amenability and property P of ... more In this paper we study the relationship between amenability, inner amenability and property P of a von Neumann algebra. We give necessary conditions on a locally compact group G to have an inner invariant mean m suchthat m(V) = 0 for some compact neighborhood F of G invariant under the inner automorphisms. We also give a sufficient condition on G (satisfied by the free group on two generators or an I.C.C. discrete group with Kazhdan's property T, e.g., SL(«, Z), n > 3) such that each linear form on L (G) which is invariant under the inner automorphisms is continuous. A characterization of inner amenability in terms of a fixed point property for left Banach G-modules is also obtained. Amenable locally compact groups and [IN]-groups are inner amenable. The group G is [IN] if there exists a compact neighborhood V of the identity e in G such that a~xVa = V for all a e G. Furthermore when G is connected, then G is amenable if and only if G is inner amenable (see [17]). A recent account of amenability is given in [21]. Let J' be a von Neumann algebra on a Hilbert space H and let Jü' be the commutant of ./#. For T e 38(H) (the space of bounded linear operators on H), let CT be the weak*-closed convex subset of 38'(H) generated by {U*TU ; U e Jfu}, where ^u is the group of unitary elements in Jf. (Note that 38(H) has a unique predual [28, p. 47].) Jf is said to have property P if CT xx Jf' # 0 for each T e 38(H). Let VN((j) denote the von Neumann algebra on L (G) generated by {lx;x e G} where lxh(t) = h(xt), t e G. A well-known result of Schwartz [29] asserts
In his study of amenable unitary representations, M. E. B. Bekka asked if there is an analogue fo... more In his study of amenable unitary representations, M. E. B. Bekka asked if there is an analogue for such representations of the remarkable fixed-point property for amenable groups. In this paper, we prove such a fixed-point theorem in the more general context of a Gamenable von Neumann algebra M , where G is a locally compact group acting on M. The Følner conditions of Connes and Bekka are extended to the case where M is semifinite and admits a faithful, semifinite, normal trace which is invariant under the action of G.
Proceedings of the Edinburgh Mathematical Society, 1989
In [5], Ky Fan proved the following remarkable amenability “invariant subspace” theorem:Let G be ... more In [5], Ky Fan proved the following remarkable amenability “invariant subspace” theorem:Let G be an amenable group of continuous, invertible linear operators acting on a locally convex space E. Let H be a closed subspace of finite codimension n in E and X⊂E be such that:(i) H and X are G-invariant;(ii) (e + H) ∩X is compact convex for all e ∈ E;(iii) X contains an n-dimensional subspace V of E. Then there exists an n-dimensional subspace of E contained in X and invariant under G.
Proceedings of the American Mathematical Society, 1988
Let G G be a locally compact group and p ∈ [ 1 , ∞ ] p \in [1,\infty ] . Let π p {\pi _p} be the ... more Let G G be a locally compact group and p ∈ [ 1 , ∞ ] p \in [1,\infty ] . Let π p {\pi _p} be the isometric representation of G G on L p ( G ) {L_p}(G) given by π p ( x ) f ( t ) = f ( x − 1 t x ) Δ ( x ) 1 / p {\pi _p}(x)f(t) = f({x^{ - 1}}tx)\Delta {(x)^{1/p}} . Let A p ′ {\mathcal {A}’_p} be the commutant of A p {\mathcal {A}_p} in B ( L p ( G ) ) B({L_p}(G)) . In this paper we determine those G G for which: (*) A p ′ {\mathcal {A}’_p} contains a nonzero compact operator. We prove, among other things, that if p ∈ [ 1 , ∞ ) p \in [1,\infty ) , then (*) holds if and only if G ∈ [ I N ] G \in [IN] , and that if p = ∞ p = \infty , then (*) holds if and only if G G is inner amenable.
Proceedings of the American Mathematical Society, 1986
The purpose of this note is to prove that if G G is an amenable locally compact noncompact group,... more The purpose of this note is to prove that if G G is an amenable locally compact noncompact group, then the set of topological left invariant means on L ∞ ( G ) {L_\infty }(G) has cardinality 2 2 d {2^{{2^d}}} , where d d is the smallest cardinality of the covering of G G by compact sets. We also prove that in this case the spectrum of the bounded left uniformly continuous complex-valued functions contains exactly 2 2 d {2^{{2^d}}} minimal closed invariant subsets (or left ideals)
and New South Wales. Responsibility for any errors remains with the author. 2 Institutional indep... more and New South Wales. Responsibility for any errors remains with the author. 2 Institutional independence refers to the autonomy of the Legal Aid Authority from its sponsoring Ministry in terms of its location inside or outside Government. Operational independence refers to the LAA"s autonomy from Government (and other influences) in relation to the granting or refusing of legal aid. Financial independence refers to the LAA"s autonomy from others in relation to its funding. 3 The Irish LAB is technically an agency of the MOJ but it is situated outside the MOJ. 4 LA Ontario is described as an operational agency of the Government of Ontario but it is outside the Government. 5 Described as a Crown Corporation which is not a Government agency. 6 This gives the Government the important protection of deniabilitythe decision was not taken by us, but by an independent body, the LAA.
Of the four Consultation Papers produced by the Department for Constitutional Affairs over the su... more Of the four Consultation Papers produced by the Department for Constitutional Affairs over the summer of 2003, in many ways the most interesting was Constitutional Reform: A Supreme Court for the United Kingdom, not just for what it did say, but also for what it did not. For example, respondents were not asked whether the Government should replace the House of Lords with a Supreme Court or not. That was taken as a given. Yet the omission was all the more curious in the light of the fact, as subsequently became clear, that at least half of the current Law Lords do not favour the introduction of a Supreme Court.
International Journal of the Legal Profession, 1994
The legal aid system in England and Wales is in a stage of rapid change. Faced with a burgeoning ... more The legal aid system in England and Wales is in a stage of rapid change. Faced with a burgeoning budget, the Lord Chancellor's Department has decided to reduce eligibility, create standard fees for criminal cases and set a standard rate for Civil Legal Aid work. The Legal Aid Board's (LAB) new'franchising'scheme also has management of costs as a concern, 2 but the professed aim is broader: it is to ensure a'value for money'service through the accreditation of legal aid practices that meet standards for quality. This presages ...
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Papers by Alan Paterson