On a regularity condition for group algebras of non abelian locally compact groups

A Banach algebra $A$ is said to be a zero Jordan product determined Banach algebra if, for every Banach space $X$ , every bilinear map $\unicode[STIX]{x1D711}:A\times A\rightarrow X$ satisfying $\unicode[STIX]{x1D711}(a,b)=0$ whenever $a$ , $b\in A$ are such that $ab+ba=0$ , is of the form $\unicode[STIX]{x1D711}(a,b)=\unicode[STIX]{x1D70E}(ab+ba)$ for some continuous linear map $\unicode[STIX]{x1D70E}$ . We show that all $C^{\ast }$ -algebras and all group algebras $L^{1}(G)$ of amenable locally compact groups have this property and also discuss some applications.

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Small isomorphisms between group algebras

Glasgow Mathematical Journal ◽

10.1017/s0017089500007990 ◽

1991 ◽

Vol 33 (1) ◽

pp. 21-28 ◽

Cited By ~ 4

Author(s):

G. V. Wood

Keyword(s):

Group Algebras ◽

Locally Compact Groups ◽

Compact Groups ◽

Locally Compact ◽

Algebra Isomorphism

If G1 and G2 are locally compact groups and the algebras Ll(G1) and Ll(G2) are isometrically isomorphic, then G1 and G2 are isomorphic (Wendel, 1952, [8]). There is evidence that the following generalization of Wendel's result is true.If T is an algebra isomorphism of L1(G1) onto L1(G2) with ∥T∥ < √2, then G1, and G2 are isomorphic.

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Orthogonality preserving linear maps on group algebras

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004115000110 ◽

2015 ◽

Vol 158 (3) ◽

pp. 493-504 ◽

Cited By ~ 9

Author(s):

J. ALAMINOS ◽

J. EXTREMERA ◽

A. R. VILLENA

Keyword(s):

Locally Compact Group ◽

Group Algebras ◽

Locally Compact Groups ◽

Compact Groups ◽

Locally Compact ◽

Orthogonal Functions ◽

Orthogonality Conditions ◽

Linear Maps ◽

Jordan Homomorphisms ◽

The One

AbstractWe consider several types of orthogonality conditions on the group algebra L1(G) of a locally compact group G such as f$\ast $g = 0, f$\ast $g☆ = 0, f☆$\ast $g = 0, f$\ast $g = g$\ast $f = 0 and f$\ast $g☆ = g☆$\ast $f = 0, and we describe the linear maps Φ: L1(G) → L1(H) between the group algebras of locally compact groups G and H that take orthogonal functions of L1(G) into orthogonal functions of L1(H). Roughly speaking, they are weighted homomorphisms in the case where we are concerned with the one-sided orthogonality conditions and weighted Jordan homomorphisms in the case where we treat the two-sided orthogonality conditions.

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Isomorphisms between the second duals of group algebras of locally compact groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100074491 ◽

1996 ◽

Vol 119 (4) ◽

pp. 657-663 ◽

Cited By ~ 1

Author(s):

Hamid-Reza Farhadi

Keyword(s):

Compact Group ◽

Group Algebra ◽

Locally Compact Group ◽

Group Algebras ◽

Locally Compact Groups ◽

Compact Groups ◽

Locally Compact ◽

Algebra Isomorphism ◽

Continuous Automorphism

AbstractLet G be a locally compact group and L1(G) be the group algebra of G. We show that G is abelian or compact if every continuous automorphism of L1(G)** maps L1(G) onto L1(G) This characterizes all groups with this property and answers a question raised by F. Ghahramani and A. T. Lau in [7]. We also show that if G is a compact group and θ is any (algebra) isomorphism from L1(G)** onto L1(H)**, then H is compact and θ maps L1(G) onto L1(H).

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