ALGEBRA STRUCTURE

FROM TENSOR’S RESULTS ROOM

AND THE MATRIX SPACE OF THE OPERATOR’S ROOM IS LIMITED

IN THE HILBERT ROOM

Sadiq Khan (033019)

Mathematics Study Program

FPMIPA

UPI Bandung

**Abstract**. Suppose H is Hilbert’s room, a Hilbert n-tuple space (H^{n)}is Hilbert’s space as well. By identifying each matrix in Mn(B(H)) as a limited operator in H^{n it can} be indicated that Mn(B(H)) is a subaljabar* of B(H^{n).} Furthermore, it can be shown that there is a canonical isomorphism* of Mn(B(H)) in B(Hn) which causes the structure of the subaljabar* Mn(B(H)) to be similar to the structure of the operator algebra B(H^{n).} Through canonical isomorphism* we define norm in Mn(B(H)) in such a way as the norm in B(H^{n).} With the same algebraic and norm structure, Mn(B(H)) can be seen as algebra C* as well as B(H^{n).} This problem will be even more interesting if Mn(B(H)) is seen as the resulting space tensor square matrix space of the complex space with algebra operator B(H) (notified space Mm,n ⊗ B(H)), then hilbert n-tuple space (H^{n)}is seen as the result of tensor of n-tuple complex space with Hilbert H space (notified C^{n} ⊗ H).

Keywords: Algebra-C*, Algebra Operator, Hilbert n-tuple Room, Limited Operator, Canonical Isomorphism*, Subaljabar*, Tensor Multiplication Room.

1. Introduction

Various approaches can be used to study operator algebra, among them through the approach of algebraic and analytical properties of the matrix over linear space. A n×n matrix over linear space R can be seen as a linear transformation from an n-tuple linear space into the same n-tuple linear space, thus the n× n matrix is the linear operator of the n-tuple vector space. Furthermore a n×n matrix over the limited operator space B(H) can be viewed as the operator in the Hilbert n-tuple space.

An interesting problem to discuss when the matrix is seen as a linear combination of tensor element i i b =Σα ⊗b, with i b ∈ B(H), and i α scalar matrix n × n element from Mn (matrix space over complex space). So if the matrix space is treated as the operator’s room in hilbert space, then the result may also be treated the same, as if the matrix space is an algebra-C*.

BIBLIOGRAPHY

[1] Anton, H. (1987). *Linear Elementer Algebra, Fifth Edition. (Translation by Pantur Silaban)*. Jakarta: Erlangga.

[2] Bartle, Robert G., & Sherbert, Donald R. (2000). *Introduction to Real Analysis, Third Edition.* New York: John Wiley & Sons.

[3] Churchill, Ruel V., & Brown, James W. (1990). *Complex Variable and Application, Fifth Edition.* New York: McGraw-Hill.

[4] Conway, J. B. (1990). ). *Graduate Texts in Mathematics, A Course in Functional analysis.* New York: Springer-Verlag.

[5] Durbin, J. R. (1992). *Modern Algebra an Introduction.* New York: John Wiley and Sons.

[6] Effros, E. G., & Ruan, Z. J. (2000). *Space Operator.* London: Oxford University Press.

[7] Hungerford, Thomas W. (1974). *Graduate Texts in Mathematics, Algebra*. New York: Springer-Verlag.

[8] Lipschutz, Seymour. (1989). *3000 Solved Problems in Linear Algebra*. New York: McGraw-Hill.

[9] Murphy, G. J. (1990). *C*-Algebra and Operator Theory.* Sand Diego: Academic Press.

[10] Roman, S. (1992). *Graduate Texts in Mathematics, Advanced Linear Algebra.* New York: Springer-Verlag.

[11] Rosjanuardi, R. (2000). *“Tensor Products”.* Presentation Material at FMIPA ITB Doctoral Program, Bandung.

[12] Rosjanuardi, R., Sumiaty, E., & Muhtar, S. (2006). *Algebra Operators and Quantum Mechanics.* Bandung: Indonesian Education University.

[13] Sumartono, Harry. (2005). *Algebra-C* Group of Abelian Discrete Groups.* Thesis on the FPMIPA Mathematics Study Program UPI, Bandung: not published.

[14] Smith, Larry. (1998). *Linear Algebra, Third Edition.* New York: Springer-Verlag.

[15] Wahyudin. (2000). *Introduction to Abstract Algebra.* Bandung: Delta Bawean.

[16] Wikipedia. (2007). *Tensor Product*. [Online]. Available: http://en.wikipedia.org/wiki/Tensor_product [17 April 2007]
[17] Wikipedia. (2007). *Dual Space*. [Online]. Available: http://en.wikipedia.org/wiki/Dual_space [8 Mei 2007]